LAUSR

The aim of this work is to study the existence of a periodic solution for some nondensely nonautonomous partial functional differential equations with infinite delay in Banach spaces. We assume that the linear part is not necessarily densely defined and generates an evolution family. We use Massera’s approach (Duke Math 17:457–475, 1950), we prove that the existence of a bounded solution on $$\mathbb {R}^{+}$$R+ implies the existence of an $$\omega $$ω-periodic solution. In nonlinear case, we use a fixed point for multivalued maps to show the existence of a periodic solution. Finally, we consider a reaction diffusion equation with delay to illustrate the main results of this work.